Limiting distributions of curves under geodesic flow on hyperbolic manifolds

Abstract

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a hyperbolic $n$-manifold of finite volume under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold.

Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1({\mathbb{H}}^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial {\mathbb{H}}^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in ${\mathbb{H}}^n$ covering the closed immersed submanifold.

In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial {\mathbb{H}}^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure.

The proof uses dynamical properties of unipotent flows on homogeneous spaces of $\mathrm{SO}(n,1)$ of finite volume